Problem: Multiply the following complex numbers, marked as blue dots on the graph: $(4 e^{2\pi i / 3}) \cdot (2 e^{\pi i / 12})$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $4 e^{2\pi i / 3}$ ) has angle $\frac{2}{3}\pi$ and radius $4$ The second number ( $2 e^{\pi i / 12}$ ) has angle $\frac{1}{12}\pi$ and radius $2$ The radius of the result will be $4 \cdot 2$ , which is $8$ The angle of the result is $\frac{2}{3}\pi + \frac{1}{12}\pi = \frac{3}{4}\pi$ The radius of the result is $8$ and the angle of the result is $\frac{3}{4}\pi$.